Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function.
A) ; saddle point:
B) ; relative maximum value:
C) ; relative minimum value:
D) there are no critical points

Question

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. ​  ​
A) ; saddle point: 
B) ; relative maximum value: 
C) ; relative minimum value: 
D) there are no critical points

Quizplus · Accepted Answer

The critical point can be found by setting the derivative equal to zero:

f'(x) = 3x^2 - 6x = 0

Solving for x, we get x = 0 or x = 2.

To classify the nature of each critical point, we need to take the second derivative:

f''(x) = 6x - 6

Plugging in x = 0, we get f''(0) = -6, which means the point x = 0 is a relative maximum value.

Plugging in x = 2, we get f''(2) = 6, which means the point x = 2 is a relative minimum value.

Therefore, the answer is C.

Find the Critical Point(s) of the Function